Finite Group Databasebeta

← #697121

Subgroups — #697121

Order 512.

2728 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3C2×#2347C211yesmaximal, characteristic, complemented
4C2×#2317C211yesmaximal, characteristic, complemented
5#16678C212yesmaximal, complemented
6#16676C212yesmaximal, complemented
7#15301C214yesmaximal
8#15298C214yesmaximal, complemented
9#15137C212yesmaximal, complemented
10#15131C212yesmaximal, complemented
11#14870C212yesmaximal, complemented
12#14869C212yesmaximal, complemented
13#14264C211yesmaximal, characteristic, complemented
14#14006C212yesmaximal, complemented
15#13904C211yesmaximal, characteristic, complemented
16#13875C212yesmaximal, complemented
17#13839C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2487—28no—
23#2485—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#398C2214yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#394C2212yes—
30#2347—28no—
31#2317—28no—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2211yescharacteristic
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#387C2212yes—
42C2×#386C2212yes—
43C2×#385C2214yes—
44C2×#385C2212yes—
45C2×#384C2212yes—
46C2×#383C2212yes—
47C2×#382C2212yes—
48C2×#381C2214yes—
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2211yescharacteristic
58#1445C2212yes—
59#1445C2212yes—
60#1444C2212yes—
61#1444C2214yes—
62#1443C2214yes—
63#1442C2212yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2211yescharacteristic
67#1441C2211yescharacteristic
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2212yes—
73#1440C2214yes—
74#1439C2211yescharacteristic
75#1439C2211yescharacteristic
76#1439C2214yes—
77#1439C2214yes—
78#1438C2212yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86C4×#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—28no—
98C2×#117D418yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2314yes—
136#400—28no—
137#399—28no—
138#398—48no—
139#397—28no—
140#396—48no—
141#394—28no—
142#389—28no—
143#389—24no—
144#389—24no—
145#389—28no—
146#389—28no—
147#389—28no—
148#389—28no—
149#388—216no—
150#388—28no—
151#387—28no—
152#386—28no—
153#385—216no—
154#385—28no—
155#384—28no—
156#383—28no—
157#382—28no—
158#381—48no—
159#380—24no—
160#380—24no—
161#380—24no—
162#380—28no—
163#380D418yes—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—28no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—28no—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C25C2411yescharacteristic
196C25C2411yescharacteristic
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—41no—
203C22×D4—41no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4#4114yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4×D4—416no—
235#117—232no—
236#117—232no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4C2×D418yes—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—28no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D418yes—
332C2×C42#4118yes—
333#95—22no—
334#95—22no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—42no—
354#95—44no—
355#95—42no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95C2×D418yes—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—21no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C2×#4112yes—
381C242−1+412yes—
382C24C2511yescharacteristic
383C2×D4—28no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—88no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4#121116yes—
481C22×C4C22×D414yes—
482C22×C4C22×D418yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C42−1+412yes—
495C22×C42−1+414yes—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—416no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C4⋊C4—88no—
511C4⋊C4—84no—
512C4⋊C4—88no—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—88no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—216no—
526C42—216no—
527C42—44no—
528C42—48no—
529C42—88no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C2×#12114yes—
576C23C23×D411yescharacteristic
577C23C23×D411yescharacteristic
578C23C22×#4111yescharacteristic
579C23C22×#4112yes—
580C23C2×H(5,𝔽2)12yes—
581C23C2×2−1+412yes—
582C23C2×2−1+411yescharacteristic
583C23C2×2−1+411yescharacteristic
584C23#58512yes—
585C23#58511yescharacteristic
586C23#58511yescharacteristic
587D4—416no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—232no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—816no—
675C2×C4—88no—
676C2×C4C2×#12114yes—
677C2×C4C2×#12114yes—
678C2×C4#52514yes—
679C2×C4#53614yes—
680C2×C4#54718yes—
681C2×C4#55218yes—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—44no—
694C22—42no—
695C22—44no—
696C22—42no—
697C22—42no—
698C22—82no—
699C22—82no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22C22×#12111yescharacteristic
705C22C2×#52511yescharacteristic
706C22C2×#53611yescharacteristic
707C22#346012yes—
708C22C2×#54712yes—
709C22C2×#55212yes—
710C22#348511yescharacteristic
711C22#348611yescharacteristic
712C22#348812yes—
713C22#348911yescharacteristic
714C22#348911yescharacteristic
715C22#349412yes—
716C22#349512yes—
717C22#349511yescharacteristic
718C22#349511yescharacteristic
719C22#349811yescharacteristic
720C22#350011yescharacteristic
721C22#351112yes—
722C22#351212yes—
723C22#355212yes—
724C22#355512yes—
725C22#355912yes—
726C22#356512yes—
727C4—24no—
728C4—44no—
729C4—42no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—44no—
735C4—82no—
736C4—82no—
737C4—81no—
738C4—81no—
739C4—84no—
740C4—82no—
741C4#241318yes—
742C2—44no—
743C2—82no—
744C2C2×#241311yescharacteristic
745C2#3525611yescharacteristic
746C2#3563311yescharacteristic
747C2#3608011yescharacteristic
748C2#3608711yescharacteristic
749C2#3793812yes—
750C2#3954612yes—
751C2#4291411yescharacteristic
752C2#4291511yescharacteristic
753C2#4513512yes—
754C2#4533312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2347C2semidirect
4. C2×#2317C2semidirect
5. #16678C2semidirect
6. #16676C2semidirect
8. #15298C2semidirect
9. #15137C2semidirect
10. #15131C2semidirect
11. #14870C2semidirect
12. #14869C2semidirect
13. #14264C2semidirect
14. #14006C2semidirect
15. #13904C2semidirect
16. #13875C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#24852
C2×#24872
C2×#23472
C2×#23172
#166782
#166762
#153012
#152982
#151372
#151312
#148702
#148692
#142642
#140062
#139042
#138752
#138392